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Binomial Probability Calculator

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When something is tried a fixed number of times, each attempt independent and each with the same chance of working, the binomial distribution gives the odds of any number of successes. Enter the trials, the rate and the count, and get every form of the answer at once.

0.05 and 5% both work

Try:

How to use the binomial probability calculator

  1. 1Enter the number of trials and the probability of success on any one of them.
  2. 2Enter the number of successes you are asking about.
  3. 3Read all five probabilities — exactly, fewer than, at most, more than and at least.
  4. 4Use the distribution chart to see where your value sits against the whole range.

Examples

Ten trials at even odds

Input
n = 10, p = 0.5, k = 3
Result
Exactly 3: 11.72%. At most 3: 17.19%. At least 3: 94.53%. Mean 5, standard deviation 1.58, most likely outcome 5.

A quality check

Input
200 units sampled, a 2% defect rate, asking about 8 or more defects
Result
The probability of seeing at least that many when the true rate really is 2% — the figure that says whether a batch is unusual or merely unlucky.

About the binomial probability calculator

Where the formula comes from

The probability of one particular sequence with k successes among n trials is the success probability multiplied by itself k times, times the failure probability n minus k times. That much is just independence. The remaining piece is counting how many distinct sequences have k successes in them, which is the binomial coefficient — n choose k.

Putting them together gives the whole distribution. Exactly three successes in ten fair trials happens 120 different ways, each with probability one in 1,024, giving 11.72%. Every other figure on the page is a sum of terms of that same shape.

Independence is the assumption that usually breaks

The arithmetic is elementary and the modelling is where things go wrong, almost always at independence. Trials are independent when the result of one tells you nothing about the next, and real processes are full of quiet violations: a machine that drifts out of tolerance over a shift, an operator who learns, a batch of raw material shared between units.

The signature of the problem is real data spread wider than the distribution says it should be — successes clustering rather than scattering. When that shows up, the fix is not a different calculator but a different model, one that lets the underlying rate vary. The binomial is a good description of a stable process and a poor one of a drifting one.

Frequently asked questions

When does the binomial distribution not apply?
When any of its four conditions fails. The number of trials must be fixed in advance, each trial must have two outcomes, the probability must be the same every time, and the trials must be independent. Sampling without replacement from a small population breaks the third and fourth at once — each draw changes the odds for the next — and calls for the hypergeometric distribution instead.
What is the difference between 'at most 3' and 'fewer than 3'?
Whether 3 itself is included. At most 3 covers 0, 1, 2 and 3; fewer than 3 covers 0, 1 and 2. The gap between them is exactly the probability of landing on 3, which here is 11.72%, and that is far from negligible. Both are shown separately because misreading one for the other is the most frequent error in using this distribution.
How does it stay accurate with large numbers of trials?
By working in logarithms. The binomial coefficient for 500 trials is a number with over a hundred digits, and multiplying it by a probability small enough to compensate overflows to infinity times zero on the way. Computing the log of the coefficient with a log-gamma function and exponentiating only at the end keeps every intermediate value in range.
What is the normal approximation shown alongside?
For large n the binomial's shape approaches a normal curve, and before computers that approximation was how these problems were solved. It is displayed with a continuity correction when it is reasonable to use — the usual rule being at least ten expected successes and ten expected failures — and marked as unreliable when it is not. The exact figure above it is the one to quote; the approximation is there for comparison.
Does a low probability mean something is wrong?
Not by itself. Every specific outcome is unlikely when there are many of them: exactly 50 heads in 100 tosses has only an 8% chance, and it is the single most likely result. The question worth asking is about the tail — how likely a result this extreme or more so — which is what the 'at least' and 'at most' figures give you.

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